Flory-Rehner equations

Table 21.1 shows the recipe for obtaining NR specimen without filler. The specimen was cured at 145°C for 30 min. The cross-link density was measured by allowing it to swell in toluene and calculated to be 1.47 X 10 mol cm using the Flory-Rehner equation. The average number of... 

As pointed out earlier, an elastomer cross-linked above its gel point will not dissolve in a solvent, but will absorb it and swell. The swelling will continue until the forces of swelling balance the retroactive forces of the extended chains of the network. The cross-link density can then be calculated from the degree of swelling using the Flory-Rehner equation ... 

Crosslink density of PU films. The PU films were subjected to swelling tests in dimethylformamide. The crosslink densities of the films were thereafter estimated from equilibrium swelling data using a modified version of the Flory-Rehner equation. The swelling tests as well as the calculation of crosslink density are described in detail in a previous paper (6). 

Other relationships between % and an observable physical property such as osmotic pressure [20, 43], freezing point depression of polymer [20, 52] or solvent [20, 53], and gas liquid chromatography [46-54], were established in like fashion. The relationship determined for swelling of cross-linked polymer to thermodynamic equilibrium in excess liquid has particular significance for the subject of this review. It is given here in the form of the Flory-Rehner equation. 

An important methodology established early on is an equilibrium volume swelling approach and application of the Flory-Rehner equation [134, 135]. While this is satisfactory for simple one component elastomer networks it is not particularly convenient experimentally and is of dubious value for multi-component elastomers. 

Since vr0jvT varies only slowly with vr0, while v is a rather sensitive function of the inverse swelling ratio, the result of substituting vr into the Flory-Rehner equation is a vapp which is too large, but which still reflects rather faithfully any changes in vr0 brought about by varying the network chain density. Thus we may write as an approximation... 

While one of the < 2 values is equal to the peak position of envelope I, the other value cannot be determined accurately from Figure 1 due to the uncertainty of the peak position and low intensity of the broad peak (envelope II) Yeo has thus calculated (3) the interaction (X) parameter based on the modified Flory-Rehner equation (16) ... 

The Flory-Rehner equation (2,13) was used to interpret the swelling of the polyurethane networks in pure dioxane in order to calculate, the molecular weight between crosslinks. Here, we set qQ= 1 and x the network-solvent interaction parameter, constant and assumed equal to 0.35 (3). Networks with degrees of swelling (q ) of 6.41, 11 and 21.1 yielded Mc values of 7,700,... 

It is remarkable, that the prediction of a volume phase transition of nonionic polymer gels was done by Dusek and Patterson by discussing the Flory-Rehner equation (Dusek and Patterson 1968)). They predicted that a strong concentration dependence of x facilitates the occurrence of phase transition in polymer gels. The phase transition should be analogous to the coil-globule transition of a single polymer chain. 

The equilibrium degree of swelling or the gel volume of shows a large change in response to external conditions. Only when the volume change is discontinuously and the coexistence of two phases is experimentally proofed, we should use the term phase transition . By applying the stability criteria on the Flory-Rehner equation one can calculate the conditions of phase transition (binodal and spinodal curve, critical point). For details see e.g. (Onuki 1993). 

Evaluating the other two derivatives occurring in Eq. (3.125) after substitution for AGmix and AGei from Eqs. (3.120) and (3.121), respectively, we obtain (cf. Eq. (3.48)) the Flory-Rehner equation (Flory and Rehner, 1943,1950) ... 

Another use of swelling tests on rubbers quite unrelated to chemical resistance is to estimate the degree of cure, higher swelling indicating a lower cross-link density. The cross-link density is estimated from the Flory-Rehner equation ... 

The heat of mixing of polymer and swelling agent, which may be positive, negative or zero. The Flory-Rehner equation may be written... 

The important point is that the original Flory-Rehner equation contains no thermoelastic front factor term, yet simple analogy with the mechanical equations shows that one must exist, even for unfilled elastomers. The need becomes more obvious on considering reinforced elastomers, where equilibrium swelling is greatly reduced. In an empirical manner, equation (10.6) may be modified to... 

The eorrelation between thermodynamic parameters and the value of an equilibrium swelling is given by Flory-Rehner equation used now in a modified form ... 

If the Xi23 value is considered as a criterion of existence of a maximum of equilibrium swelling of polymer in the mixed solvent, a maximum of Q should correspond to the minimum of %i23. For practical use of Eq. [6.2.2] it is necessary to know parameters 5 13,5 23, and Xn- The values X13 and X23 can be determined by the Flory-Rehner equation, with data on swelling of a crosslinked elastomer in individual solvents 1 and 2. The evaluation of the Xn value can be carried out with use of results of the experimental evaluation of vapor pressure, viscosity and other characteristics of a binary mixture. ... 

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